torsdag 6 december 2012

Liberation from the Prandtl Spell 2

Continuing the argument in the preceding post, we consider a (laminar) shear layer L between to regions of fluid with different velocity. If epsilon is the viscosity, the width of the shear layer then scales like square-root-of-epsilon = epsilon^0.5 and the velocity gradient like 1/epsilon^0.5, which gives a total dissipation

D = integral-over-L epsilon x grad u^2 ds ~ A x epsilon^0.5

where A is the area of L. If the area is bounded, which is the case for a boundary layer of a bluff body, then D tends to zero with epsilon, which indicates a vanishing effect with vanishing viscosity, contrary to Prandtl's basic postulate.

On the other hand, the length of the rolls of 3d rotational separation stretching into the flow behind the body may increase with decreasing viscosity thus keeping D positive under vanishing viscosity. The effect of the rolls on the boundary of a bluff body and total dissipation, which determine drag and lift, would thus stay the same under vanishing viscosity, while the rolls after the body would get longer.

In the case of a turbulent shear layer, D ~ A x epsilon^0.2 according to experimental data, but the conclusion would be similar.